You run a single-item sealed bid auction where you sell an old camera. There are three bidders and the value of the camera for each of them is described by a certain (known) random variable: With probability 0.9 the value is 100+x where x is taken uniformly at random from the interval [-1,1]. With probability 0.1 the value is 300+x where x is as before. The 300 value represents the case that the item has a special nostalgic value for the buyer.

The values of the camera to the three bidders are thus i.i.d random variables. (The reason for adding this small random x is to avoid ties, and you can replace the interval [-1,1] with [-ε, ε] for a small ε, if you wish.) If you don’t want to worry about ties you can simply think about the value being 100 with probability 0.9 and 300 wit probability 0.1.

### The basic question

The basic questions for you the seller is:

**What is the highest expected revenue you, the seller, can guarantee and what is your optimal allocation policy.**

I’d like to see the answer for this question. But let me challenge your intuition with two simpler questions.

1) Can the seller guarantee an expected revenue of 120 or more?

2) What (roughly) is the optimal allocation policy

a) Highest bidder wins.

b) Highest bidder wins if his bid passes some reserve price.

c) The item is allocated at random to the bidders with probabilities depending on their bids.

### Background: Myerson’s paper and his revenue equivalence theorem

The most relevant paper to this post is a well-known paper *Optimal auction design* by Roger Myerson. Myerson considered the case of a single-item sealed-bid auction where the bidders’ values for the item are independent identical random variable.

Note that** **I** **did not specify** **the price that the winning bidder is going to pay for the camera. The reason is that according to a famous theorem by Myerson (referred to as the *revenue equivalence theorem*), when the bidders are strategic, the expected revenues for the seller are determined by the allocation rule and are independent from the pricing policy! (Well, you need to assume a reasonable pricing policy…)

For example, if the item is allocated to the highest bidder then the expected price will be the second highest price. If the price is determined by the second highest bid (Vickery’s auction) then each bidder has no incentive to give a bid which is different from its value. But even if the item will be allocated to the first bidder for the highest price, the expected revenues will still be the same! When you analyze an auction mechanism like ours you can assume that the payments are set in a way that the bidders have no incentive not to bid the true value the camera has. This is sometimes referred to as the *revelation principle*.

Myerson considered a mechanism which sometimes lead to higher revenues compared to allocating the item to the highest bidder: The seller set a * reserve price *and the item is allocated to the highest bidder if the bid passes this reserve price, and is not allocated at all otherwise. In the paper Myerson also considered more complicated allocation rules which are important in the analysis where the item is allocated to bidders according to some probabilities based on their bids.

### Polls

This time we have two questions and two polls:

Once again this is a game-theory question. I hope it will lead to interesting discussion like the earlier one on tic-tac-toe.

### A little more Background: Auctions and Vickery’s second price auction.

(From Wikipedia) An **auction** is a process of buying and selling goods or services by offering them up for bid, taking bids, and then selling the item to the highest bidder. In economic theory, an auction may refer to any mechanism or set of trading rules for exchange.

In our case, we consider an auction of a single item (the camera) and each bidder is giving a sealed bid.

(Again from Wikipedea) A **Vickrey auction** is a type of sealed-bid auction, where bidders submit written bids without knowing the bid of the other people in the auction, and in which the highest bidder wins, but the price paid is the second-highest bid. The auction was first described academically by Columbia University professor William Vickrey in 1961 though it had been used by stamp collectors since 1893.^{[2]} This type of auction is strategically similar to an English auction, and gives bidders an incentive to bid their true value.

Josué OrtegaI will read it carefully later, but I must say that I enjoy your posts a lot. They cover many subfields and the intuition questions are quite interesting. Cheers from Vienna!

gilIf I can attach probabilities as I want to who gets the camera according to the bids, including determinstic probablities, then, Isn’t option C a special case of options A+B? Option A by giving a probability of 1 to who posted the highest bid, and option B by adding 0 probablity below the reserve price. If I can give such probablities, I think I can reach 136.

Gil KalaiPost authorYes, option 2 is more general than option 1 and option 3 is more general than option 2. The question is what is the maximum revenue that can be guaranteed and my Myerson theorem we can assume that the price makes the allocation rule truthful.

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